Spread options, Farkas's lemma and linear programming

Vladimir Piterbarg derives necessary and sufficient conditions for the existence of a joint distribution consistent with given marginals and the distribution of the spread in terms of no-arbitrage conditions among certain payouts. He also proposes a generic numerical approach to constructing such distributions, and identifying payouts that realise arbitrage if it exists

Options on individual underlyings are very liquid in a variety of markets. In many markets, moreover, options on linear combinations of underlyings are also reasonably liquid. Of primary interest to us are markets with liquid spread options (that is, options on the difference of two underlyings), such as constant maturity swap (CMS) spread options in interest rate markets. Our discussion also naturally extends to other important examples such as foreign exchange markets with cross-rate options